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A finite extension $L/K$ is normal when every irreducible polynomial in $K[T]$ having one root in $L$ splits completely over $L$.

Suppose first that $L/K$ is normal. Choose ${\alpha }_1,\dots ,{\alpha }_m$ with $L=K({\alpha }_1,\dots ,{\alpha }_m)$, and let $f_j$ be the minimal polynomial of ${\alpha }_j$ over $K$. Every $f_j$ splits in $L$ by normality, and $L$ is generated by their roots because it is already generated by the particular roots ${\alpha }_j$. Thus $L$ is the splitting field of $f_1\cdots f_m$.

Conversely, suppose that $L$ is the splitting field of $f\in K[T]$. Every $K$-embedding $\sigma :L\to \overline{K}$ permutes the roots of $f$, so $\sigma (L)=L$. By the embedding characterization of normality, $L/K$ is normal. This proves the finite normal extension as a splitting field criterion.

True. If $M/K$ is normal, the finite normal extension as a splitting field criterion writes $M$ as the splitting field over $K$ of some $f\in K[T]$. Since $K\subseteq L\subseteq M$, the same field $M$ is the splitting field of $f$, now regarded as a polynomial over $L$. Hence $M/L$ is normal.